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Non-monotonic Reasoning

Non-monotonic Reasoning. Are we having a pop quiz today? You assume not. But can you prove it? In commonsense reasoning, we often jump to conclusions, can’t always list the assumptions we made, need to retract conclusions, when we get more information.

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Non-monotonic Reasoning

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  1. Non-monotonic Reasoning • Are we having a pop quiz today? • You assume not. • But can you prove it? • In commonsense reasoning, • we often jump to conclusions, • can’t always list the assumptions we made, • need to retract conclusions, when we get more information. • In first-order logic, our conclusion set is monotonically growing.

  2. The Closed World Assumption • KB contains: Student(Joe), Student(Mary) • Query: Student(Fred)? • Intuitively, no; but can’t prove it. • Solution: when appropriate, close the predicate Student. • X Student(X) <=>X=Joe v X=Mary • Closing can be subtle when multiple predicates are involved: • X In(X) <=> Out(X)

  3. More on CWA • Negation as failure: x,y,z edge(x,z)  path(z,y)  path(x,y) x,y edge(x,y)  path(x,y) edge(A,B), edge(B,C), edge(A,D) • Conclude: path(C,D). • Domain-closure assumption: the only named constants in the KB exist in the universe. • Unique-names assumption: every constant is mapped to a different object in the universe. (already assumed in Description Logics and Databases).

  4. Default Rules Bird(X) C(Flies(X)) : Flies(X) is consistent. Flies(X) • Application of default rules: the order matters! Liberal(X) Hunter(X) C(Dem(X)) C(Rep(X)) Dem(X) Rep(X)  X  (Dem(X)  Rep(X)) Liberal(Tom), Hunter(Tom)

  5. Minimal Models: Circumscription • Consider only models in which the extension of some predicates is minimized. •  X (Bird(X) abnormal(X))  Flies(X) • Some predicates are distinguished as “abnormal”. • An interpretation I1 is preferred to I2 if: • I1 and I2 agree on the extensions of all objects, functions and non-abnormal predicates. • The extension of abnormal in I1 is a strict subset of its extension in I2. • KB |= S, if S is satisfied in every minimal model of KB (I is minimal if no I2 is preferred to it).

  6. But Uncertainty is Everywhere • Medical knowledge in logic? • Toothache <=> Cavity • Problems • Too many exceptions to any logical rule • Hard to code accurate rules, hard to use them. • Doctors have no complete theory for the domain • Don’t know the state of a given patient state • Uncertainty is ubiquitous in any problem-solving domain (except maybe puzzles) • Agent has degree of belief, not certain knowledge

  7. Ways to Represent Uncertainty • Disjunction • If information is correct but complete, your knowledge might be of the form • I am in either s3, or s19, or s55 • If I am in s3 and execute a15 I will transition either to s92 or s63 • What we can’t represent • There is very unlikely to be a full fuel drum at the depot this time of day • When I execute pickup(?Obj) I am almost always holding the object afterwards • The smoke alarm tells me there’s a fire in my kitchen, but sometimes it’s wrong

  8. Numerical Repr of Uncertainty • Interval-based methods • .4 <= prob(p) <= .6 • Fuzzy methods • D(tall(john)) = 0.8 • Certainty Factors • Used in MYCIN expert system • Probability Theory • Where do numeric probabilities come from? • Two interpretations of probabilistic statements: • Frequentist: based on observing a set of similar events. • Subjective probabilities: a person’s degree of belief in a proposition.

  9. KR with Probabilities • Our knowledge about the world is a distribution of the form prob(s), for sS. (S is the set of all states) • s S,0  prob(s)  1 • sSprob(s) = 1 • For subsets S1 and S2, prob(S1S2) = prob(S1) + prob(S2) - prob(S1S2) • Note we can equivalently talk about propositions:prob(p  q) = prob(p) + prob(q) - prob(p  q) • where prob(p) means sS | p holds in s prob(s) • prob(TRUE) = 1 • prob(FALSE) = 0

  10. Probability As “Softened Logic” • “Statements of fact” • Prob(TB) = .06 • Soft rules • TB  cough • Prob(cough | TB) = 0.9 • (Causative versus diagnostic rules) • Prob(cough | TB) = 0.9 • Prob(TB | cough) = 0.05 • Probabilities allow us to reason about • Possibly inaccurate observations • Omitted qualifications to our rules that are (either epistemological or practically) necessary

  11. Probabilistic Knowledge Representation and Updating • Prior probabilities: • Prob(TB) (probability that population as a whole, or population under observation, has the disease) • Conditional probabilities: • Prob(TB | cough) • updated belief in TB given a symptom • Prob(TB | test=neg) • updated belief based on possibly imperfect sensor • Prob(“TB tomorrow” | “treatment today”) • reasoning about a treatment (action) • The basic update: • Prob(H)  Prob(H|E1)  Prob(H|E1, E2)  ...

  12. Ache Ache Cavity 0.04 0.06 0.01 0.89 Cavity Basics • Random variable takes values • Cavity: yes or no • Joint Probability Distribution • Unconditional probability (“prior probability”) • P(A) • P(Cavity) = 0.1 • Conditional Probability • P(A|B) • P(Cavity | Toothache) = 0.8

  13. Bayes Rule • P(B|A) = P(A|B)P(B) ----------------- P(A) A = red spots B = measles We know P(A|B), but want P(B|A).

  14. C A P Prob F F F 0.534 F F T 0.356 F T F 0.006 F T T 0.004 T F F 0.048 T F T 0.012 T T F 0.032 T T T 0.008 Conditional Independence • “A and P are independent” • P(A) = P(A | P) and P(P) = P(P | A) • Can determine directly from JPD • Powerful, but rare(I.e. not true here) • “A and P are independent given C” • P(A|P,C) = P(A|C) and P(P|C) = P(P|A,C) • Still powerful, and also common • E.g. suppose • Cavities causes aches • Cavities causes probe to catch Ache Cavity Probe

  15. C A P Prob F F F 0.534 F F T 0.356 F T F 0.006 F T T 0.004 T F F 0.012 T F T 0.048 T T F 0.008 T T T 0.032 Conditional Independence • “A and P are independent given C” • P(A | P,C) = P(A | C) and also P(P | A,C) = P(P | C)

  16. Suppose C=True P(A|P,C) = 0.032/(0.032+0.048) = 0.032/0.080 = 0.4

  17. P(A|C) = 0.032+0.008/ (0.048+0.012+0.032+0.008) = 0.04 / 0.1 = 0.4

  18. Summary so Far • Bayesian updating • Probabilities as degree of belief (subjective) • Belief updating by conditioning • Prob(H)  Prob(H|E1)  Prob(H|E1, E2)  ... • Basic form of Bayes’ rule • Prob(H | E) = Prob(E | H) P(H) / Prob(E) • Conditional independence • Knowing the value of Cavity renders Probe Catching probabilistically independent of Ache • General form of this relationship: knowing the values of all the variables in some separator set S renders the variables in set A independent of the variables in B. Prob(A|B,S) = Prob(A|S) • Graphical Representation...

  19. Computational Models for Probabilistic Reasoning • What we want • a “probabilistic knowledge base” where domain knowledge is represented by propositions, unconditional, and conditional probabilities • an inference engine that will computeProb(formula | “all evidence collected so far”) • Problems • elicitation: what parameters do we need to ensure a complete and consistent knowledge base? • computation: how do we compute the probabilities efficiently? • Belief nets (“Bayes nets”) = Answer (to both problems) • a representation that makes structure (dependencies and independencies) explicit

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